Solvability of nonlinear elliptic equations with gradient terms

Patricio Felmer (1) , Alexander Quaas (2) , Boyan Sirakov (3)

1 DIM - Departamento de Ingenieria Matematica and Centro de Modelamiento Matematico
2 Departamento de Matematica [Valparaiso]
3 Departamento de Matematica

We study the solvability in the whole Euclidean space of coercive quasi-linear and fully nonlinear elliptic equations modeled on $\Delta u\pm g(|\nabla u|)= f(u)$, $u\ge0$, where $f$ and $g$ are increasing continuous functions. We give conditions on $f$ and $g$ which guarantee the availability or the absence of positive solutions of such equations in $\R^N$. Our results considerably improve the existing ones and are sharp or close to sharp in the model cases. In particular, we completely characterize the solvability of such equations when $f$ and $g$ have power growth at infinity. We also derive a solvability statement for coercive equations in general form.

Domaines: Equations aux dérivées partielles [math.AP]

Patricio Felmer, Alexander Quaas, Boyan Sirakov. Solvability of nonlinear elliptic equations with gradient terms. Journal of Differential Equations, 2013, 254 (11), pp.4327-4346. ⟨hal-00741161v2⟩